The mathematical know-how in polymake is bundled in so called applications. Applications are software components that can be installed and used independently of each other; however, they are allowed to contain functions that "cross the boundaries".

The list below is aimed to give you just the first idea about the standard applications shipped with polymake. Choose an application you are interested in via the menu box in the navigation pane; in the tutorial you will find a lot of examples of what can be done with an application, and the following topics lead you to the reference part describing all user-visible components.

Finally, there are collections of external links to other useful information sources on the corresponding research field.

polytope

This is the historically first application, and the largest one.

It deals with convex pointed polyhedra. It allows to define a polyhedron either as a convex hull of a point set, an intersection of halfspaces, or as an incidence matrix without any embedding. Then you can ask for a plenty of its (especially combinatorial) properties, construct new polyhedra by modifying it, or study the behavior of the objective functions.

There is a wide range of visualization methods for polyhedra, even for dimensions > 4 and purely combinatorial descriptions, including interfaces to interactive geometry viewers (such as JavaView or geomview), generating PostScript drawings and povray scene files.

topaz

topology application zoo deals with abstract simplicial complexes. A complex is given as a list of facets. You can ask for its global properties (manifold recognition, homology groups, etc.), explore the local vertex environment (stars, links, etc.), and make a lot of constructions.

The visualization means are constrained, as they are mostly based on the graph (1-skeleton) of a complex.

surface

This is the beginning of an application treating polyhedral surfaces, that is, closed polyhedral 2-manifolds. For the start, the application provides some basic combinatorial properties and visualization.